Madrid, January 17, 21 1 Andreas Schmitt Institut für Theoretische Physik Technische Universität Wien 1 Vienna, Austria Sound modes and the two-stream instability in relativistic superfluids M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, PRD 87, 651 (213) M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, arxiv:131.5953 [hep-ph] A. Schmitt, arxiv:1312.5993 [hep-ph] two-fluid picture of a superfluid role reversal in first and second sound two-stream instability
Madrid, January 17, 21 2 Superfluid hydrodynamics: relevance for compact stars r-mode instability pulsar glitches precession asteroseismology superfluid turbulence (?) Cas A, Chandra X-Ray Observatory Superfluidity in dense matter Nuclear matter Quark matter neutrons (T c 1 kev) color-flavor locked phase (T c 1 MeV) hyperons color-spin locked phase (T c 1 kev)
Madrid, January 17, 21 3 Two-fluid picture of a superfluid (liquid helium) London, Tisza (1938); Landau (191) relativistic: Khalatnikov, Lebedev (1982); Carter (1989) superfluid component : condensate, carries no entropy normal component : excitations (Goldstone mode), carries entropy ε p phonon roton p Hydrodynamic eqs. two sound modes 1st sound in-phase oscillation (primarily) density wave 2nd sound out-of-phase oscillation (primarily) entropy wave
Madrid, January 17, 21 First and second sound in non-relativistic systems liquid helium K.R. Atkins et al. (1953) ultracold fermionic gas (exp.) L.A. Sidorenkov et al., Nature 98, 78 (213) weakly interacting Bose gas H.Hu, et al., New Journ.Phys. 12, 3 (21) unitary Fermi gas E. Taylor et al., PRA 8, 5361 (29)
Madrid, January 17, 21 5 Goals How does the two-fluid picture arise from a microscopic field theory? M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, PRD 87, 651 (213) Compute sound modes in a relativistic superfluid (and in the presence of a superflow) M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, arxiv:131.5953 [hep-ph] A. Schmitt, arxiv:1312.5993 [hep-ph]
Madrid, January 17, 21 6 Lagrangian and superfluid velocity starting point: complex scalar field L = ( ϕ) 2 m 2 ϕ 2 λ ϕ Bose condensate ϕ = ρ e iψ spontaneously breaks U(1) zero temperature: single-fluid system Field theory current j µ ( ψ) 2 stress-energy tensor T µν Hydrodynamics λ µ ψ nv µ g µν L + ( ψ)2 λ µ ψ ν ψ (ɛ + P )v µ v ν g µν P superfluid velocity v µ = µ ψ µ µ = ψ
Madrid, January 17, 21 7 Relativistic two-fluid formalism (page 1/2) write stress-energy tensor as T µν = g µν Ψ + j µ ν ψ + s µ Θ ν generalized pressure Ψ: Ψ = P in superfluid and normal-fluid rest frames, Ψ depends on momenta µ ψ, Θ µ Ψ = Ψ[( ψ) 2, Θ 2, ψ Θ] generalized energy density Λ Ψ + j ψ + s Θ Λ is Legendre transform of Ψ, Λ depends on currents j µ, s µ Λ = Λ[j 2, s 2, j s]
Madrid, January 17, 21 8 Relativistic two-fluid formalism (page 2/2) j µ = s µ = Ψ ( µ ψ) = B µ ψ + A Θ µ Ψ Θ µ = A µ ψ + C Θ µ B = 2 Ψ ( ψ), 2 A = Ψ ( ψ Θ) C = 2 Ψ Θ 2 entrainment coefficient compute A, B, C from microscopic physics,, Μ 2 Λ 2.5 2. 1.5 1..5...2..6.8 all temperatures 1 6 Μ 2 Λ 3 Entrainment coefficient 2PI T 3 T 5 2 1..1.2.3. (very) small temperatures
Madrid, January 17, 21 9 Microscopic calculation for arbitrary T (page 1/2) effective action density in the 2PI formalism (CJT) Γ[ρ, S] = U(ρ) 1 2 Tr ln S 1 1 2 Tr[S 1 (ρ)s 1] V 2[ρ, S] V 2 [ρ, S]: two-loop two-particle irreducible (2PI) diagrams use Hartree approximation impose Goldstone theorem by hand solve self-consistency equations for condensate ρ and M, δm
Madrid, January 17, 21 1 Microscopic calculation for arbitrary T (page 2/2) microscopic calculation done in normal-fluid rest frame identify effective action density with generalized pressure Γ[µ, T, ψ] = Ψ restrict to weak coupling no dependence on renormalization scale consider uniform superflow v neglect dissipation thermodynamics with (µ, T, v) compute entrainment coefficient, sound velocities etc.
Madrid, January 17, 21 11 Results I: critical velocity v.5 3.29.3 instability at v = v c negative energies in Goldstone dispersion ɛ k (v) < Εk Μ.2.1 T T.5 T c v T T c v. 1..5..5 generalization to Landau s original argument ɛ k k v < v.8.6..2 uniform superfluid non superfluid...2..6.8 1. 1.2 1. dashed line: without backreaction of condensate shaded region: dissipation, turbulence? c similar phase diagram for holographic superfluid I. Amado, D. Arean, A. Jimenez-Alba, K. Landsteiner, L. Melgar and I. S. Landea, arxiv:137.81 [hep-th]
Madrid, January 17, 21 12 Results II: sound speeds and mixing angle ultra-relativistic (towards) non-relativistic.6.5.6.5 sound speed u..3.2.1..3.2.1.5.52.5.8.6.. Α 1 mixing angle Α Π Π Α 1 pure Μ wave Α 2 pure T wave Π Π Α 2 Π 2..2..6.8 Π 2..2..6.8 α = arctan δt δµ role reversal in first and second sound!
Madrid, January 17, 21 13 Sound speeds and mixing angle with superflow ultra-relativistic (towards) non-relativistic superflow coupling.7.6.6.5.5 u..3 parallel anti parallel..3.2.2.1.1.. Α 1 Α 1 Π Π Α Α 2 Α 2 Π Π Π 2..2..6.8 Π 2..2..6.8
Madrid, January 17, 21 13 Sound speeds and mixing angle with superflow ultra-relativistic (towards) non-relativistic superflow coupling.7.6.6.5.5.. u.3.3.2.2.1.1.. Α 1 Α 1 Π Π Α Α 2 Α 2 Π Π Π 2..2..6.8 Π 2..2..6.8
Madrid, January 17, 21 13 Sound speeds and mixing angle with superflow ultra-relativistic (towards) non-relativistic superflow coupling.7.6.6.5..5. u.3.2.1.3.2.1.. Α 1 Α 1 Π Π Α Α 2 Α 2 Π Π Π 2..2..6.8 Π 2..2..6.8
Madrid, January 17, 21 13 Sound speeds and mixing angle with superflow ultra-relativistic (towards) non-relativistic superflow coupling.7.6.6.5..5. u.3.2.1.3.2.1.. Α 1 Α 1 Π Π Α Α 2 Α 2 Π Π Π 2..2..6.8 Π 2..2..6.8
Madrid, January 17, 21 1 Results III: two-stream instability compute sound speed close to Landau s critical velocity v.8.6..2 uniform superfluid non superfluid...2..6.8 1. 1.2 1. c.8 T.T c Θ Π.5 T.T c Θ Π Re u.6. Im u..2.5..988.989.99.991.992.993.99.995 6.1.988.989.99.991.992.993.99.995 complex sound speeds one mode damped, one mode explodes plasma physics: O. Buneman, Phys.Rev. 115, 53 (1959); D.T. Farley, PRL 1, 279 (1963) general two-fluid system: L. Samuelsson, C. S. Lopez-Monsalvo, N. Andersson, G. L. Comer, Gen. Rel. Grav. 2, 13 (21) relevance for superfluids: N. Andersson, G. L. Comer, R. Prix, MNRAS 35, 11 (2)
Madrid, January 17, 21 15 All directions v v.5v c T v.967v c T.5.5.5 1..5.5 1..5.5 1..5.5.5.5.5.5.5.5 1..5.5 1..5.5 1..5.5.5.5.5 (superflow pointing to the right)
Madrid, January 17, 21 16 Instability window in phase diagram.5.995 u 1 v..3.2.1.68.62 uniform superfluid...2..6.8 v vc T.99.985 uniform superfluid Im u.98.975..2..6.8 c tiny window for weak coupling λ =.5 (varying λ shows that the window grows with λ) region with u > 1: problem in the formalism? (Hartree? enforced Goldstone theorem?) very small T : qualitatively different angular structure of instability
Madrid, January 17, 21 17 Summary a superfluid is a two-fluid system, and this can be derived from microscopic physics the two sound modes in a (weakly coupled, relativistic) superfluid can reverse their roles (in terms of density and entropy waves) at large relative velocities of the two fluids, there is a dynamical instability ( two-stream instability )
Madrid, January 17, 21 18 Outlook start from fermionic theory D. Müller, A. Schmitt, work in progress behavior beyond critical velocity sound modes (role reversal): predictions for He or ultracold gases? apply to compact stars neutron superfluid & ion lattice: N. Chamel, D. Page and S. Reddy, PRC 87, 3583 (213) two-stream instability: instability more prominent at strong coupling? holographic approach: C.P.Herzog and A.Yarom, PRD 8, 162 (29); I.Amado, D.Arean, A.Jimenez-Alba, K.Landsteiner, L.Melgar, I.S.Landea, arxiv:137.81 [hep-th] time evolution of instability I. Hawke, G. L. Comer and N. Andersson, Class. Quant. Grav. 3, 157 (213) relevance for compact stars, e.g., pulsar glitches N. Andersson, G. L. Comer, R. Prix, MNRAS 35, 11 (2)